Homotopy analysis method (HAM) is applied to obtain the approximate solution of inner-resonance of tangent cushioning packaging system based on critical components. The solution is obtained in the form of infinite series with components which can be easily calculated. Using a convergence-control parameter, the HAM utilizes a simple method to adjust and control the convergence region of the infinite series solution. The obtained results show that the HAM is a very accurate technique to obtain the approximate solution.
1. Introduction
One of the most important subjects in packaging system is to investigate products damaged due to being dropped. Many researchers have investigated cushioning packaging system in this special field [1, 2]. The mechanical and electronic products are composed of large number of elements and, generally, damage at the so-called critical components. To prevent any damage, a critical component and a cushioning packaging are included in a package system [3]. The following assumptions are made basically in the last decade by the researchers [1, 4].
The researchers considered that the packaging system is a spring-mass, single degree of freedom system.
The use of simple linear or nonlinear springs for cushioning packaging may not be appropriate.
Wang et al. [2] considered a linear model for this system, while the oscillation in the package system is inborn nonlinearity (see [4, 5]). Our goal of this paper is to obtain the approximate solution of inner-resonance of tangent nonlinear cushioning packaging system with critical components introduced in [1, 4] using HAM, which is one of the semilinear approximate analytical methods.
In the last two decades, many researchers have employed the approximate analytical methods such as adomian decomposition method (ADM), variational iteration method (VIM), homotopy perturbation method (HPM), and HAM to solve differential equations. These methods give the solution of the differential equations in the form of infinite series. One of the advantages of approximate analytical methods is that these methods do not produce rounding-off errors. Contrary to the implicit finite difference method (FDM), the approximate analytical methods do not require the numerical solution of the systems of differential equations.
We apply the HAM to construct the series solution for the inner-resonance of tangent nonlinear cushioning packaging system with critical components. An advantage of HAM over perturbation methods is that it is not dependent on small or large parameters. As it is well known, perturbation methods cannot be applied to all nonlinear equations because these methods are based on the existence of small or large parameters. Besides, nonperturbation methods are independent of small parameters. According to [6], both of the two techniques (perturbation and nonperturbative techniques) cannot provide a simple procedure to adjust or control the convergence region and rate of given approximate series. According to [7, 8], HAM also allows for fine-tuning of convergence region and rate of convergence by allowing an auxiliary parameter ħ to vary. To the best of our knowledge, this is the first attempt at solving the inner-resonance of tangent nonlinear cushioning packaging system with components approximately using the HAM.
In recent years, the HAM has been successfully applied for solving various nonlinear systems of equations in many branches of mathematics and sciences, such as strongly coupled reaction-diffusion system [9], fractional Lorenz system [7], coupled Schrodinger-KdV equation [10], Burgers and coupled Burgers equation [11], system of second-order BVPs [8], and HIV infection CD4+T-cell [12]. For more studies of HAM and its applications, the readers are also referred to see [13–16].
Our paper is organized as follows.
In Section 2, we introduce mathematical modelling of the inner-resonance of tangent nonlinear cushioning packaging system with critical components. In Section 3, we present a description of the HAM on system of equations, as expanded by previous researchers in particular [9, 12], applied to the inner-resonance of tangent nonlinear cushioning packaging system with critical components. We prove the convergence of homotopy series solution for the inner-resonance of tangent nonlinear cushioning packaging system with critical components also in this section. In Section 4, we have applied the HAM to obtain the approximate solution of inner-resonance of tangent nonlinear cushioning packaging system with critical components. Finally, in Section 5, we give the conclusion of this study.
2. Modeling and Equations
Generally, imagine that everything we know and have a relationship to, including things such as art, clothes, possessions, homes, gardens, trees and fields, mountains, lakes, oceans, continents, our friends, and loved ones, are instances where resonance can and does occur. Now, consider that everything we do and think is an attempt to seek out and to return to the experience of resonance, a return to the feeling of Belonging and feeling that things feel right. Even though we may not identify the motivation for making and forming certain relationships, the real attraction and value of any relationship is whether it is fulfilling and makes us feel good. We are searching for the feeling of resonance. The profound nature of these kinds of experiences is dependent on the nature and quality of the relationship we can have to something of an external nature, but we also have the potential to experience resonance within our own body and our inner being, and this I would call a “state of Inner Resonance.” When a practitioner consciously perceives resonance happening in a therapeutic relationship between themselves and patient, it is also a state of Inner Resonance. In this circumstance, the practitioners create the appropriate environment, conditions, and quality of presence so that they can receive the patients intention to be understood and received. In another way, the patient’s need to experience resonance (whether consciously or unconsciously) is met. Inner Resonance, then, is a state of receptive presence or conscious empathy between two or more people.
In Figure 1, the model of packaging system with critical component is shown to be considered as a nonlinear spring with stiffness coefficientk2. It can also idealize the joining part between the mass of critical componentm1and the main part of the productm2as a linear spring with stiffness coefficientk1. According to Figures 1 and 2, the motion of this system can be written as [1, 4]
(1)m1d2x(t)dt2+k1(x(t)-y(t))=0,m2d2y(t)dt2+2k2dbπtan(π2dby(t))-k1(x(t)-y(t))=0,
with initial conditions
(2)x(0)=0,x′(0)=2gh,y(0)=0,y′(0)=2gh.
The model of packaging system with critical component (m1≪m2) andx1=x(t),x2=y(t).
Theħ-curves ofX′(0.1)andY′(0.1)obtained by the 5-order approximation of HAM.
Table 1 summarize the meanings of parameters and variables. To simplify (1), new variables are introduced as
(3)X=xL,Y=yL,T=tT0,
where
(4)T0=m2k2,L=2dbπ.
We define the frequency parameters of the critical component and main part of product asw1=k1/m1andw2=k2/m2, respectively. The notationsλ1=w1/w2andλ2=m1/m2are considered as parameter ratio and mass ratio, respectively. By considering all parameters defined, (1) can be equivalently written in the following system of nonlinear equations [1]:
(5)d2XdT2+w012X-w012Y=0,d2YdT2+w022Y+13Y3+215Y5+(1-w022)X=0,
with initial conditions
(6)X(0)=0,X′(0)=T0L2gh,Y(0)=0,Y′(0)=T0L2gh,
whereX=X(T),Y=Y(T)and
(7)w01=λ1,w02=1+λ12λ2.
List of variables and parameters (modified from [1, 4]).
Parameters and variables
Illustration
x
Displacement response of critical component
y
Main body of the product
m1
Mass of critical component
m2
Main part of product
k1
Stiffness coefficient
k2
Stiffness coefficient
db
Compression limit of the cushioning pad
h
Dropping height
g
Gravity acceleration
2gh
Dropping shock velocity of the product
3. Homotopy Analysis Method (HAM)
To apply the HAM, the nonlinear system (5) is considered. We make initial gusses onX(T)andY(T)such that they satisfy the initial conditions (6) that are defined as
(8)X(0)=X0=T0L2ghT,Y(0)=Y0=T0L2ghT.
The auxiliary linear operators ℒX and ℒY are selected as
(9)ℒX=d2XdT2,ℒY=d2YdT2,
satisfying the following properties:
(10)ℒX(c1,XT+c2,X)=0,ℒY(c1,YT+c2,Y)=0,
where c1,X, c2,X, c1,Y, and c2,Y are integral constants. Define the homotopy maps
(11)ℋX(X^(T;q),Y^(T;q))0000=(1-q)ℒX[X^(T;q)-X0(T)]000000-qħHX(t)NX[X^(T;q),Y^(T;q)],ℋY(X^(T;q),Y^(T;q))0000=(1-q)ℒY[Y^(T;q)-Y0(T)]000000-qħHY(t)NY[X^(T;q),Y^(T;q)],
where q∈[0,1] is an embedding parameter, ħ is nonzero auxiliary parameter, HX and HY are auxiliary functions, and NX and NY are nonlinear operators that are defined as
(12)NX[X^(T;q),Y^(T;q)]=∂2X^(T;q)∂T2+w012X^(T;q)-w012Y^(T;q),NY[X^(T;q),Y^(T;q)]=∂2Y^(T;q)∂T2+w022Y^(T;q)+(Y^(T;q))33+215(Y^(T;q))5+(1-w022)X^(T;q).
Clearly, whenq=0, we have the homotopy maps
(13)ℋX(X^(T;0),Y^(T;0))=ℒX[X^(T;0)-X0(T)],ℋY(X^(T;0),Y^(T;0))=ℒY[Y^(T;0)-Y0(T)].
And whenq=1, we have
(14)ℋX(X^(T;1),Y^(T;1))0000=-ħHX(T)NX[X^(T;1),Y^(T;1)],ℋY(X^(T;1),Y^(T;1))0000=-ħHY(T)NY[X^(T;1),Y^(T;1)].
Thus, by requiring
(15)ℋX(X^(T;q),Y^(T;q))=ℋY(X^(T;q),Y^(T;q))=0,
we can obtain
(16)(1-q)ℒX[X^(T;q)-X0(T)]0000=qħHX(T)NX[X^(T;q),Y^(T;q)],(1-q)ℒY[Y^(T;q)-Y0(T)]0000=qħHY(T)NY[X^(T;q),Y^(T;q)].
Ifq=0andq=1, the homotopy equations are as follows:
(17)X^(T;0)=X0,X^(T;1)=X(T),Y^(T;0)=Y0,Y^(T;1)=Y(T).
Asqvaries from0to1, the solution of the nonlinear system (5) will vary from the initial guessesX0(T)andY0(T)to the exact solutionsX(T)andY(T)of the nonlinear system (5). ExpandingX^(T;q)andY^(T;q)as a Taylor series with respect toqyields
(18)X^(T;q)=X0+∑m=1∞Xmqm,Y^(T;q)=Y0+∑m=1∞Ymqm,
where
(19)Xm=1m!∂mX^(T;q)∂qm|q=0,Ym=1m!∂mY^(T;q)∂qm|q=0.
According to [17], the convergence of the series (18) strongly depends on the auxiliary parameter ħ. Note that if q=1, then
(20)X^(T;1)=X=X0+∑m=1∞Xm,Y^(T;1)=Y=Y0+∑m=1∞Ym.
According to definitions (18), the governing equations for the unknowns can be deduced from the zeroth-deformation equations (16). For further analysis, the vectors are defined as
(21)X^n={X0,X1,…,Xn},Y^n={Y0,Y1,…,Yn}.
Differentiating (16) m-times with respect to q, dividing by m!, and setting q=0 give the linear equations
(22)ℒX[Xm-χmXm-1]=ħHX(T)Rm,X(X→m-1,Y→m-1),ℒY[Ym-χmYm-1]=ħHY(T)Rm,Y(X→m-1,Y→m-1),
with initial conditions
(23)Xm(0)=0,Xm′(0)=0,Ym(0)=0,Ym′(0)=0,
where
(24)Rm,X(X→m-1,Y→m-1)=Xm-1′′+w012Xm-1-w012Ym-1,(25)Rm,Y(X→m-1,Y→m-1)=Ym-1′′+w022Ym-1+13∑i1=0m-1Yi1∑i2=0m-1-i1Yi2Ym-1-i1-i2+215∑i1=0m-1Yi1∑i2=0m-1-i1Yi2∑i3=0m-1-i1-i2Yi3×∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i4+(1-w022)Xm-1,(26)χm∶={0m≤11m>1.
UsingHX(T)=HY(T)=1, the solution of them-order deformation equations (22) form≥1becomes
(27)Xm=χmXm-1+ħ∬Rm,X(X→m-1(τ),Y→m-1(τ))dτdτ+c1,XT+c2,X,Ym=χmYm-1+ħ∬Rm,Y(X→m-1(τ),Y→m-1(τ))dτdτ+c1,YT+c2,Y.
The coefficients c1,X, c2,X, c1,Y, and c2,Y are determined using initial conditions (23).
3.1. Convergence TheoremTheorem 1.
The seriesX(T)=X0+∑m=1∞XmandY(T)=Y0+∑m=1∞Ymconverge whereXmandYmare governed by (22) under definitions (24)–(26); X and Y must be the solutions of system of (5).
Proof.
If the series∑m=0∞Xmand∑m=0∞Ymare convergent, we can write
(28)SX=∑m=0∞Xm,SY=∑m=0∞Ym.
And it holds that
(29)limm→∞Xm=limm→∞Ym=0.
From (22) and using (9), we have
(30)ħ∑m=1∞Rm,X(X→m-1,Y→m-1)=∑m=1∞ℒX[Xm-χmXm-1]=limm→∞∑m=0nℒX[Xm-χmXm-1]=ℒX[limm→∞∑m=0n(Xm-χmXm-1)]=ℒX[limm→∞Xn]=0,ħ∑m=1∞Rm,Y(X→m-1,Y→m-1)=∑m=1∞ℒY[Ym-χmYm-1]=limm→∞∑m=0nℒY[Ym-χmYm-1]=ℒY[limm→∞∑m=0n(Ym-χmYm-1)]=ℒY[limm→∞Yn]=0.
Sinceħ≠0, then
(31)∑m=1∞Rm,X(X→m-1,Y→m-1)=0,(32)∑m=1∞Rm,Y(X→m-1,Y→m-1)=0.
Substituting (24) into (31) and simplifying it, we obtain
(33)∑m=1∞Rm,X(X→m-1,Y→m-1)0000=∑m=1∞(Xm-1′′+w012Xm-1-w012Ym-1)0000=∑m=1∞Xm-1′′+w012∑m=1∞Xm-1-w012∑m=1∞Ym-10000=d2dT2∑m=0∞Xm+w012∑m=0∞Xm-w012∑m=0∞Ym0000=X′′+w012X-w012Y=0.
We repeat this process and substitute (25) into (32), and simplifying it, we obtain
(34)∑m=1∞Rm,Y(X→m-1,Y→m-1)0000=∑m=1∞Ym-1′′+w022∑m=1∞Ym-1000000+∑m=1∞(1-w022)Xm-1000000+13∑m=1∞[∑i1=0m-1Yi1∑i2=0m-1-i1Yi2Ym-1-i1-i2]000000+215∑m=1∞[∑i1=0m-1Yi1∑i2=0m-1-i1Yi2∑i3=0m-1-i1-i2Yi3000000000000000×∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i4].
For the first three terms of (34), we can easily conclude that
(35)∑m=1∞Ym-1′′+w022∑m=1∞Ym-1+∑m=1∞(1-w022)Xm-10000=Y′′(T)+w022Y(T)+(1-w022)X(T).
For the fourth and fifth terms in (34), we have
(36)13∑m=1∞[∑i1=0m-1Yi1∑i2=0m-1-i1Yi2Ym-1-i1-i2]00=13∑m=1∞∑i1=0m-1Yi1∑i2=0m-1-i1Yi2Ym-1-i1-i200=13∑i1=0∞∑m=i1+1∞Yi1∑i2=0m-1-i1Yi2Ym-1-i1-i200=13∑i1=0∞∑j=0∞Yi1∑i2=0jYi2Yj-i200=13∑i1=0∞Yi1∑j=0∞∑i2=0jYi2Yj-i200=13∑i1=0∞Yi1∑i2=0∞∑j=i2∞Yi2Yj-i200=13∑i1=0∞Yi1∑i2=0∞Yi2∑j=i2∞Yj-i200=13(∑m=0∞Ym)3=13Y3(T),215∑m=1∞[∑i1=0m-1Yi1∑i2=0m-1-i1Yi2∑i3=0m-1-i1-i2Yi3000000×∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i4]00=215∑m=1∞∑i1=0m-1Yi1∑i2=0m-1-i1Yi20000×∑i3=0m-1-i1-i2Yi3∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i400=215∑i1=0∞Yi1∑m=i1+1∞∑i2=0m-1-i1Yi20000×∑i3=0m-1-i1-i2Yi3∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i400=215∑i1=0∞Yi1∑j=0∞∑i2=0jYi2∑i3=0m-1-i1-i2Yi30000×∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i400=215∑i1=0∞Yi1∑i2=0∞Yi2∑m=1+i1+i2∞∑i3=0m-1-i1-i2Yi30000×∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i400=215∑i1=0∞Yi1∑i2=0∞Yi2∑l=0∞∑i3=0lYi30000×∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i400=215∑i1=0∞Yi1∑i2=0∞Yi2∑i3=0∞Yi30000×∑m=1+i1+i2+i3∞∑i4=0m-1-i1-i2-i3Yi4Ym-1-i1-i2-i3-i400=215∑i1=0∞Yi1∑i2=0∞Yi2∑i3=0∞Yi3∑k=0∞∑i4=0kYi4Yk-i400=215∑i1=0∞Yi1∑i2=0∞Yi2∑i3=0∞Yi3∑i4=0∞∑k=i4kYi4Yk-i400=215∑i1=0∞Yi1∑i2=0∞Yi2∑i3=0∞Yi3∑i4=0∞Yi4∑k=i4kYk-i400=215Y5(T).
Thus,
(37)∑m=1∞Rm,Y(X→m-1,Y→m-1)=Y′′+w022Y+13Y3+215Y5+(1-w022)X=0.
From (2) and (23), it holds that
(38)X0=∑m=0∞Xm(0)=X0(0)+∑m=1∞Xm(0)=0,X0′=∑m=0∞Xm′(0)=X0′(0)+∑m=1∞Xm′(0)=T0L2gh,Y0=∑m=0∞Ym(0)=Y0(0)+∑m=1∞Ym(0)=0,Y0′=∑m=0∞Ym′(0)=Y0′(0)+∑m=1∞Ym′(0)=T0L2gh.
Thus,XandYsatisfy the system (5) and it must be the exact solution for (5) with the initial conditions (6).
4. Example
In this section, the HAM is applied to obtain the approximate solutions of the system (5) with the initial conditions (6). We have also used symbolic software Mathematica to solve the system of linear equations (22) with the initial conditions (23). Few components of the series solutions of (20) are given as follows:
(39)X0=T0ghπT2db,Y0=T0ghπT2db,X1=0,Y1=ħπT3ghT062db(1+gπ2T2hm220db2k20+g2π4T4h2m22210db4k22),X2=ħ2πT5ghT0k1m21202dbk2m1(1+gπ2T2hm242db2k20+g2π4T4h2m22756db4k22),Y2=ħπT3ghT062db(1+ħ+ħT22+ħT2k120k20+għπ2T4hm2k1840db2k22+gπ2T2hm220db2k20+11għπ2T4hm2840db2k2+⋯),⋮
It is clear that the HAM series solutions (20) on depend on the convergence-control parameterħwhich provides a simple way to adjust and control the convergence of the series solutions. In fact, it is very important to ensure that the series (20) are convergent. To this end, we have plottedħ-curve of X′(0.1) and Y′(0.1) by fifth- and sixth-order approximation of the HAM in Figures 2 and 3, respectively, for values m1=0.01, m2=1, k1=0.1, k2=0.02, g=0.8, h=0.01, anddb=0.9. According to these ħ-curves, it is easy to discover the valid region of convergence-control parameterħwhich corresponds to the line segment nearly parallel to the horizontal axis. For clearer presentation, these valid regions have been listed in Table 2. Furthermore, these valid regions ensure us the convergence of the obtained series. Liao [18] has pointed out that when ħ=-1, the solution obtained by the HAM is the same as the series solution obtained using HPM.
The admissible value of ħ derived from Figures 2 and 3.
m
5
6
T(t)
-1.8≤ħ≤-0.7
-1.3≤ħ≤-0.8
I(t)
-1.3≤ħ≤-0.75
-1.25≤ħ≤-0.75
Theħ-curves ofX′(0.1)andY′(0.1)obtained by the 6-order approximation of HAM.
Now, an error analysis is introduced to obtain the optimal value of convergence-control parameter ħ. Toward this end, we define φX(T;ħ) and φY(T;ħ) to be m-order approximation HAM solution as follows:
(40)φX(T;ħ)=∑j=0m-1Xj,φY(T;ħ)=∑j=0m-1Yj.
We substitute (40) into nonlinear system (5) and obtain the residual error functions ERX(X,Y,ħ1) and ERY(X,Y,ħ2) as follows:
(41)ERX(X,Y;ħ1)=d2φX(T;ħ1)dT2+w012φX(T;ħ1)-w012φY(T;ħ1),(42)ERY(X,Y;ħ2)=d2φY(T;ħ2)dT2+w022φY(T;ħ2)+13(φY(T;ħ2))3+215(φY(T;ħ2))5+(1-w022)φX(T;ħ2).
Following [19], we define the square residual error for the m-order approximation to be
(43)RX(ħ1)=∫01(ERX(X,Y,;ħ1))2dT,RY(ħ2)=∫01(ERY(X,Y,;ħ2))2dT.
We can obtain the values of ħ1 and ħ2 for which the RX(ħ1) and RY(ħ2) are minimum. The optimal values of convergence-control parameters ħ1 and ħ2 are determined by solving the system of equations as
(44)dRX(ħ1*)dħ1=0,dRY(ħ2*)dħ2=0.
In [20], several methods have been introduced to find the optimal value of ħ. In Table 3, the minimum values of RX(ħ1) and RY(ħ2) have been given with optimal values of ħ1* and ħ2* for 4-, 5-, and 6-order approximations.
The minimum values of RX(ħ*) and RY(ħ*) for various orders of approximations.
m
X(T)
Y(T)
ħ*
Minimum RX(ħ*)
ħ*
Minimum RY(ħ*)
4
−0.950213
1.60718×10-12
−0.950432
2.54142×10-8
5
−0.973770
3.80411×10-15
−0.971307
3.62018×10-11
6
−0.983485
3.84892×10-18
−0.993411
4.43599×10-14
In Table 4, the absolute errors ERX and ERY have been calculated for various T∈(0,1) when 5- and 6-order approximation HAM solutions are considered. From the table, it can be seen that the HAM provides us with the accurate approximate solution for the inner-resonance of tangent cushioning packaging system based on critical components (5).
The residual errors ERX and ERY for various T∈(0,1).
T
5-order
6-order
ERX
ERY
ERX
ERY
0.1
7.65813×10-11
1.37420×10-9
3.18762×10-13
4.47395×10-11
0.2
3.06638×10-10
2.43550×10-8
1.76197×10-14
4.84994×10-11
0.3
1.58864×10-10
8.87158×10-8
3.49043×10-12
1.21465×10-9
0.4
2.05543×10-9
5.62311×10-7
9.40057×10-12
7.36418×10-10
0.5
2.20192×10-9
1.18024×10-6
7.17227×10-11
1.31996×10-8
0.6
1.04361×10-8
6.66260×10-7
5.64075×10-11
4.10080×10-8
0.7
5.15236×10-8
3.16150×10-7
6.52645×10-10
1.44031×10-8
0.8
1.17256×10-7
1.07335×10-5
2.96157×10-9
2.22440×10-7
0.9
1.17456×10-7
1.27292×10-5
4.99250×10-9
5.66406×10-7
The residual errors ERX and ERY have been plotted in Figures 4 and 5 for T∈(0,1) and various convergence-control parametersħ. By considering these Figures, it is to be noted that the solution obtained using HAM gives an analytical solution with high order of accuracy with few iterations.
The errors of residual equation (41) using the sixth-order approximate solution for variousħandT∈(0,1).
The errors of residual equation (42) using the sixth-order approximate solution for variousħandT∈(0,1).
5. Conclusion
The homotopy analysis method (HAM) is applied to obtain approximate solution of inner-resonance of tangent cushioning packaging system based on critical components. It is shown that the HAM solution contains the convergence-control parameter ħ, which provides a simple way to adjust and control the convergence region of the resulting infinite series. The convergence of HAM is also proved for inner-resonance of tangent cushioning packaging system based on critical components. The obtained results show that HAM is an accurate and effective technique for obtaining the approximate solution of inner-resonance of tangent cushioning packaging system based on critical components.
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